Figure 1.
The schematic diagram of the moderately thick composite laminated cylindrical shell with elastic boundary conditions: (a) the whole composite shell; (b) the cross-section view of the composite shell.
Figure 1.
The schematic diagram of the moderately thick composite laminated cylindrical shell with elastic boundary conditions: (a) the whole composite shell; (b) the cross-section view of the composite shell.
Figure 2.
The modal shapes of a three-layered [0°/90°/0°] composite shell with simply-supported (S-S) boundary conditions.
Figure 2.
The modal shapes of a three-layered [0°/90°/0°] composite shell with simply-supported (S-S) boundary conditions.
Figure 3.
The modal shapes of a three-layered [0°/90°/0°] composite shell with various elastic boundary conditions.
Figure 3.
The modal shapes of a three-layered [0°/90°/0°] composite shell with various elastic boundary conditions.
Table 1.
Frequency parameters for a three-layer cross-ply cylindrical shell [0°/90°/0°] with various classical boundary conditions (R = 1 m, L/R = 5, h/R = 0.05, E2 = 1 GPa, E1/E2 = 25, μ12 = 0.25, G12 = 0.5E2, G13 = 0.5E2, G23 = 0.2E2, ρ = 1700 kg/m3; m = 1).
Table 1.
Frequency parameters for a three-layer cross-ply cylindrical shell [0°/90°/0°] with various classical boundary conditions (R = 1 m, L/R = 5, h/R = 0.05, E2 = 1 GPa, E1/E2 = 25, μ12 = 0.25, G12 = 0.5E2, G13 = 0.5E2, G23 = 0.2E2, ρ = 1700 kg/m3; m = 1).
n | WBM | Ref. [56] | Error | Ref. [57] | Error |
---|
F-F |
---|
1 | 304.179 | 304.13 | 0.02% | 304.16 | 0.01% |
2 | 26.558 | 26.58 | −0.08% | 26.56 | −0.01% |
3 | 77.027 | 74.91 | 2.83% | 74.78 | 3.01% |
4 | 144.798 | 142.93 | 1.31% | 142.51 | 1.61% |
5 | 230.986 | 229.74 | 0.54% | 228.7 | 1.00% |
| SD-SD |
1 | 151.486 | 151.49 | 0.00% | 151.49 | 0.00% |
2 | 92.564 | 92.57 | −0.01% | 92.57 | −0.01% |
3 | 95.253 | 95.37 | −0.12% | 95.27 | −0.02% |
4 | 149.999 | 150.42 | −0.28% | 150.01 | −0.01% |
5 | 232.927 | 233.97 | −0.45% | 232.94 | −0.01% |
| C-C |
1 | 159.443 | 159.31 | 0.08% | 159.44 | 0.00% |
2 | 107.889 | 107.71 | 0.17% | 107.89 | 0.00% |
3 | 108.106 | 108.05 | 0.05% | 108.11 | 0.00% |
4 | 156.945 | 157.23 | −0.18% | 156.94 | 0.00% |
5 | 236.764 | 237.7 | −0.39% | 236.76 | 0.00% |
Table 2.
Frequency parameters for two types of cross-ply composite laminated cylindrical shell with different length to radius ratios and boundary conditions (R = 1 m, h/R = 0.2, E2 = 1 GPa, E1/E2 = 40, μ12 = 0.25, G12 = 0.6E2, G13 = 0.5E2, G13 = 0.5E2, ρ = 1600 kg/m3).
Table 2.
Frequency parameters for two types of cross-ply composite laminated cylindrical shell with different length to radius ratios and boundary conditions (R = 1 m, h/R = 0.2, E2 = 1 GPa, E1/E2 = 40, μ12 = 0.25, G12 = 0.6E2, G13 = 0.5E2, G13 = 0.5E2, ρ = 1600 kg/m3).
Layer-Type | Shell Theories | S-S | S-C | C-C | C-F |
---|
L/R = 1 | L/R = 2 | L/R = 1 | L/R = 2 | L/R = 1 | L/R = 2 | L/R = 1 | L/R = 2 |
---|
[0°/90°] | HSDT [58] | 0.0804 | 0.1556 | 0.0938 | 0.1726 | 0.1085 | 0.1928 | 0.0444 | 0.0921 |
FSDT [58] | 0.0791 | 0.1552 | 0.0893 | 0.1697 | 0.1002 | 0.1876 | 0.0435 | 0.0914 |
CST [58] | 0.0866 | 0.1630 | 0.1152 | 0.1841 | 0.1048 | 0.2120 | 0.0480 | 0.0938 |
FSDT [59] | 0.0766 | 0.1519 | 0.0823 | 0.1661 | 0.0982 | 0.1737 | 0.0396 | 0.0872 |
FSDT [57] | 0.0881 | 0.1578 | 0.0921 | 0.1639 | 0.0982 | 0.1738 | 0.0396 | 0.0872 |
WBM | 0.0884 | 0.1581 | 0.0908 | 0.1631 | 0.0962 | 0.1723 | 0.0397 | 0.0873 |
[0°/90°/0°] | HSDT [58] | 0.1007 | 0.1777 | 0.1087 | 0.1972 | 0.1192 | 0.2191 | 0.0506 | 0.0995 |
FSDT [58] | 0.1004 | 0.1779 | 0.1036 | 0.1945 | 0.1093 | 0.2129 | 0.0495 | 0.0988 |
CST [58] | 0.1479 | 0.2073 | 0.1850 | 0.2662 | 0.2049 | 0.3338 | 0.0669 | 0.1099 |
FSDT [59] | 0.0996 | 0.1722 | 0.1025 | 0.1950 | 0.1083 | 0.2083 | 0.0483 | 0.0914 |
FSDT [57] | 0.0996 | 0.1726 | 0.1028 | 0.1991 | 0.1086 | 0.2084 | 0.0483 | 0.0912 |
WBM | 0.0967 | 0.1706 | 0.0993 | 0.2043 | 0.1042 | 0.2017 | 0.0472 | 0.0907 |
Table 3.
Frequency parameters for two types of cross-ply composite laminated cylindrical shells with different thickness to radius ratios under simply-supported boundary conditions (R = 1 m, L/R = 0.1, E2 = 1 GPa, E1/E2 = 40, μ12 = 0.25, G12 = 0.6E2, G13 = 0.5E2, G13 = 0.5E2, ρ = 1600 kg/m3).
Table 3.
Frequency parameters for two types of cross-ply composite laminated cylindrical shells with different thickness to radius ratios under simply-supported boundary conditions (R = 1 m, L/R = 0.1, E2 = 1 GPa, E1/E2 = 40, μ12 = 0.25, G12 = 0.6E2, G13 = 0.5E2, G13 = 0.5E2, ρ = 1600 kg/m3).
h/R | [0°/90°/90°/0°] | [90°/0°/0°/90°] |
---|
WBM | Ref. [57] | Error | Ref. [59] | Error | WBM | Ref. [57] | Error | Ref. [59] | Error |
---|
0.1 | 0.0638 | 0.0639 | −0.17% | 0.0640 | −0.32% | 0.0531 | 0.0533 | −0.38% | 0.0531 | 0.00% |
0.0656 | 0.0657 | −0.17% | 0.0657 | −0.17% | 0.0591 | 0.0592 | −0.24% | 0.0591 | −0.07% |
0.0789 | 0.0789 | −0.05% | 0.0789 | −0.05% | 0.0709 | 0.0710 | −0.14% | 0.0709 | 0.00% |
0.2 | 0.1586 | 0.1588 | −0.14% | 0.1589 | −0.20% | 0.1332 | 0.1335 | −0.22% | 0.1333 | −0.07% |
0.1676 | 0.1678 | −0.15% | 0.1683 | −0.44% | 0.1527 | 0.1528 | −0.06% | 0.1527 | 0.01% |
0.1726 | 0.1727 | −0.07% | 0.1726 | −0.01% | 0.1590 | 0.1593 | −0.18% | 0.1592 | −0.12% |
0.3 | 0.2539 | 0.2542 | −0.11% | 0.2546 | −0.27% | 0.2272 | 0.2275 | −0.12% | 0.2273 | −0.03% |
0.2669 | 0.2670 | −0.03% | 0.2669 | 0.01% | 0.2429 | 0.2430 | −0.03% | 0.2428 | 0.05% |
0.2785 | 0.2788 | −0.11% | 0.2797 | 0.43% | 0.2697 | 0.2701 | −0.14% | 0.2699 | −0.07% |
Table 4.
Frequency parameters for a three-layered cross-ply [0°/90°/0°] composite laminated cylindrical shell under simply-supported boundary conditions (E2 = 2 GPa, E1/E2 = 25, μ12 = 0.25, G12 = 0.5E2, G13 = 0.5E2, G23 = 0.2E2, ρ = 1600 kg/m3, m = 1).
Table 4.
Frequency parameters for a three-layered cross-ply [0°/90°/0°] composite laminated cylindrical shell under simply-supported boundary conditions (E2 = 2 GPa, E1/E2 = 25, μ12 = 0.25, G12 = 0.5E2, G13 = 0.5E2, G23 = 0.2E2, ρ = 1600 kg/m3, m = 1).
h/R | L/R = 1 | L/R = 2 | L/R = 3 | L/R = 4 |
---|
n = 1 | n = 2 | n = 1 | n = 2 | n = 1 | n = 2 | n = 1 | n = 2 |
---|
0.05 | 1.54962 | 1.12747 | 0.78125 | 0.51865 | 0.52133 | 0.33977 | 0.39057 | 0.25367 |
0.06 | 1.58005 | 1.18476 | 0.78545 | 0.53046 | 0.52261 | 0.34539 | 0.39112 | 0.25800 |
0.07 | 1.61120 | 1.24206 | 0.79021 | 0.54365 | 0.52408 | 0.35183 | 0.39177 | 0.26299 |
0.08 | 1.64201 | 1.29756 | 0.79546 | 0.55794 | 0.52575 | 0.35899 | 0.39250 | 0.26857 |
0.09 | 1.67168 | 1.35011 | 0.80111 | 0.57306 | 0.52760 | 0.36680 | 0.39331 | 0.27468 |
0.1 | 1.69971 | 1.39908 | 0.80707 | 0.58877 | 0.52961 | 0.37516 | 0.39421 | 0.28127 |
Table 5.
Frequency parameters for a three-layered cross-ply [0°/90°/0°] composite laminated cylindrical shell with clamped boundary conditions (E2 = 2 GPa, E1/E2 = 25, μ12 = 0.25, G12 = 0.5E2, G13 = 0.5E2, G23 = 0.2E2, ρ = 1600 kg/m3, m = 1).
Table 5.
Frequency parameters for a three-layered cross-ply [0°/90°/0°] composite laminated cylindrical shell with clamped boundary conditions (E2 = 2 GPa, E1/E2 = 25, μ12 = 0.25, G12 = 0.5E2, G13 = 0.5E2, G23 = 0.2E2, ρ = 1600 kg/m3, m = 1).
h/R | L/R = 1 | L/R = 2 | L/R = 3 | L/R = 4 |
---|
n = 1 | n = 2 | n = 1 | n = 2 | n = 1 | n = 2 | n = 1 | n = 2 |
---|
0.05 | 1.74397 | 1.45928 | 0.82781 | 0.60980 | 0.54152 | 0.37875 | 0.40177 | 0.27465 |
0.06 | 1.79223 | 1.53994 | 0.84181 | 0.63996 | 0.54739 | 0.39300 | 0.40492 | 0.28362 |
0.07 | 1.83113 | 1.60423 | 0.85551 | 0.66928 | 0.55335 | 0.40776 | 0.40814 | 0.29316 |
0.08 | 1.86213 | 1.65516 | 0.86864 | 0.69708 | 0.55934 | 0.42277 | 0.41142 | 0.30314 |
0.09 | 1.88682 | 1.69562 | 0.88099 | 0.72294 | 0.56529 | 0.43778 | 0.41473 | 0.31345 |
0.1 | 1.90656 | 1.72802 | 0.89243 | 0.74672 | 0.57113 | 0.45260 | 0.41806 | 0.32398 |
Table 6.
Frequency parameters for two types of cross-ply composite laminated cylindrical shells with classical boundary conditions (L/R = 4, h/R = 0.1, E2 = 2 GPa, E1/E2 = 25, μ12 = 0.25, G12 = 0.5E2, G13 = 0.5E2, G23 = 0.2E2, ρ = 1500 kg/m3, m = 1).
Table 6.
Frequency parameters for two types of cross-ply composite laminated cylindrical shells with classical boundary conditions (L/R = 4, h/R = 0.1, E2 = 2 GPa, E1/E2 = 25, μ12 = 0.25, G12 = 0.5E2, G13 = 0.5E2, G23 = 0.2E2, ρ = 1500 kg/m3, m = 1).
Layer- | n | SD-SD | S-S | C-C |
---|
Type | Ref. [57] | WBM | Ref. [57] | WBM | Ref. [57] | WBM |
---|
[0°/90°/0°] | 1 | 61.94 | 61.939 | 63.069 | 63.074 | 66.887 | 66.889 |
2 | 42.76 | 42.739 | 44.99 | 45.003 | 51.846 | 51.837 |
3 | 55.85 | 55.803 | 57.428 | 57.443 | 63.007 | 62.979 |
4 | 92.309 | 92.249 | 93.101 | 93.108 | 96.611 | 96.569 |
[0°/90°] | 1 | 59.523 | 59.523 | 62.065 | 62.069 | 62.677 | 62.676 |
2 | 43.199 | 43.205 | 47.847 | 47.854 | 48.488 | 48.488 |
3 | 73.147 | 73.145 | 75.891 | 75.888 | 76.138 | 76.13 |
4 | 128.58 | 128.56 | 130.02 | 130.009 | 130.13 | 130.112 |
Table 7.
Frequency parameters for two types of cross-ply composite laminated cylindrical shells with elastic boundary conditions (L/R = 4, h/R = 0.1, E2 = 2 GPa, E1/E2 = 25, μ12 = 0.25, G12 = 0.5E2, G13 = 0.5E2, G23 = 0.2E2, ρ = 1500 kg/m3, m = 1).
Table 7.
Frequency parameters for two types of cross-ply composite laminated cylindrical shells with elastic boundary conditions (L/R = 4, h/R = 0.1, E2 = 2 GPa, E1/E2 = 25, μ12 = 0.25, G12 = 0.5E2, G13 = 0.5E2, G23 = 0.2E2, ρ = 1500 kg/m3, m = 1).
Layer- | n | EC1-EC1 | EC2-EC2 | EC3-EC3 | EC4-EC4 |
---|
Type | Ref. [57] | WBM | Ref. [57] | WBM | Ref. [57] | WBM | Ref. [57] | WBM |
---|
[0°/90°/0°] | 1 | 65.844 | 65.788 | 65.767 | 62.014 | 60.544 | 65.508 | 59.906 | 58.511 |
2 | 50.056 | 50.019 | 51.223 | 51.279 | 50.187 | 49.273 | 49.672 | 49.629 |
3 | 61.725 | 61.682 | 62.916 | 62.893 | 61.619 | 59.397 | 61.709 | 61.667 |
4 | 95.949 | 95.902 | 96.592 | 96.551 | 95.548 | 102.61 | 95.949 | 95.902 |
[0°/90°] | 1 | 61.265 | 61.167 | 56.838 | 55.912 | 62.231 | 62.15 | 55.912 | 54.997 |
2 | 46.054 | 46.02 | 47.977 | 47.981 | 48.019 | 47.859 | 45.792 | 45.772 |
3 | 74.745 | 74.738 | 76.084 | 76.083 | 75.941 | 75.817 | 74.743 | 74.737 |
4 | 129.44 | 129.428 | 130.12 | 130.104 | 130.03 | 129.929 | 129.44 | 129.425 |
Table 8.
The frequency parameters for a three-layered cross-ply [0°/90°/0°] composite laminated cylindrical shell with S-elastic boundary conditions, one displacement is under elastic restrained and others are free (L/R = 4, h/R = 0.1, E2 = 2 GPa, E1/E2 = 25, μ12 = 0.25, G12 = 0.5E2, G13 = 0.5E2, G23 = 0.2E2, ρ = 1500 kg/m3).
Table 8.
The frequency parameters for a three-layered cross-ply [0°/90°/0°] composite laminated cylindrical shell with S-elastic boundary conditions, one displacement is under elastic restrained and others are free (L/R = 4, h/R = 0.1, E2 = 2 GPa, E1/E2 = 25, μ12 = 0.25, G12 = 0.5E2, G13 = 0.5E2, G23 = 0.2E2, ρ = 1500 kg/m3).
Spring Stiffness | Ku | Kv | Kw | Kϕx | Kϕθ |
---|
n = 1 | n = 2 | n = 3 | n = 1 | n = 2 | n = 3 | n = 1 | n = 2 | n = 3 | n = 1 | n = 2 | n = 3 | n = 1 | n = 2 | n = 3 |
---|
0 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 |
101 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 |
102 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 |
103 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 | 29.069 | 56.813 | 142.978 |
104 | 29.069 | 56.813 | 142.978 | 29.075 | 56.814 | 142.978 | 29.075 | 56.819 | 142.981 | 29.069 | 56.813 | 142.976 | 29.069 | 56.815 | 142.982 |
105 | 29.069 | 56.813 | 142.978 | 29.131 | 56.829 | 142.981 | 29.131 | 56.876 | 143.005 | 29.069 | 56.811 | 142.958 | 29.069 | 56.837 | 143.024 |
106 | 29.072 | 56.814 | 142.980 | 29.684 | 56.971 | 143.008 | 29.682 | 57.416 | 143.251 | 29.069 | 56.756 | 142.339 | 29.069 | 57.171 | 143.694 |
107 | 29.097 | 56.825 | 142.996 | 34.324 | 58.233 | 143.288 | 34.122 | 60.798 | 145.643 | 29.069 | 56.854 | 143.276 | 29.069 | 55.922 | 141.734 |
108 | 29.303 | 56.917 | 143.145 | 50.759 | 63.136 | 145.984 | 47.642 | 64.640 | 149.485 | 29.069 | 56.848 | 143.238 | 29.069 | 56.152 | 141.954 |
109 | 29.975 | 57.257 | 143.872 | 59.203 | 65.439 | 149.834 | 53.982 | 65.281 | 150.110 | 29.069 | 56.847 | 143.235 | 29.069 | 56.168 | 141.973 |
1010 | 30.339 | 57.469 | 144.544 | 60.257 | 65.709 | 150.405 | 54.810 | 65.347 | 150.173 | 29.069 | 56.847 | 143.235 | 29.069 | 56.170 | 141.975 |
1011 | 30.392 | 57.502 | 144.669 | 60.364 | 65.737 | 150.462 | 54.895 | 65.353 | 150.179 | 29.069 | 56.847 | 143.235 | 29.069 | 56.170 | 141.975 |
1012 | 30.398 | 57.506 | 144.683 | 60.375 | 65.739 | 150.468 | 54.904 | 65.354 | 150.180 | 29.069 | 56.847 | 143.235 | 29.069 | 56.170 | 141.975 |
Table 9.
The frequency parameters for a three-layered cross-ply [0°/90°/0°] composite laminated cylindrical shell with S-elastic boundary conditions, one displacement is under elastic restrained and others are free (L/R = 4, h/R = 0.1, E2 = 2 GPa, E1/E2 = 25, μ12 = 0.25, G12 = 0.5E2, G13 = 0.5E2, G23 = 0.2E2, ρ = 1500 kg/m3).
Table 9.
The frequency parameters for a three-layered cross-ply [0°/90°/0°] composite laminated cylindrical shell with S-elastic boundary conditions, one displacement is under elastic restrained and others are free (L/R = 4, h/R = 0.1, E2 = 2 GPa, E1/E2 = 25, μ12 = 0.25, G12 = 0.5E2, G13 = 0.5E2, G23 = 0.2E2, ρ = 1500 kg/m3).
Spring Stiffness | Ku | Kv | Kw | Kϕx | Kϕθ |
---|
n = 1 | n = 2 | n = 3 | n = 1 | n = 2 | n = 3 | n = 1 | n = 2 | n = 3 | n = 1 | n = 2 | n = 3 | n = 1 | n = 2 | n = 3 |
---|
0 | 60.837 | 66.299 | 141.103 | 59.262 | 67.658 | 141.619 | 62.532 | 67.706 | 141.578 | 62.247 | 67.373 | 141.437 | 62.686 | 67.847 | 141.630 |
101 | 60.837 | 66.299 | 141.103 | 59.262 | 67.658 | 141.619 | 62.532 | 67.706 | 141.578 | 62.247 | 67.373 | 141.437 | 62.686 | 67.847 | 141.630 |
102 | 60.837 | 66.299 | 141.103 | 59.262 | 67.658 | 141.619 | 62.532 | 67.706 | 141.578 | 62.247 | 67.373 | 141.437 | 62.686 | 67.847 | 141.630 |
103 | 60.837 | 66.299 | 141.103 | 59.262 | 67.658 | 141.619 | 62.532 | 67.706 | 141.578 | 62.247 | 67.373 | 141.437 | 62.686 | 67.847 | 141.630 |
104 | 60.837 | 66.299 | 141.103 | 59.262 | 67.658 | 141.619 | 62.532 | 67.706 | 141.578 | 62.246 | 67.371 | 141.436 | 62.686 | 67.847 | 141.630 |
105 | 60.837 | 66.300 | 141.103 | 59.263 | 67.658 | 141.619 | 62.532 | 67.707 | 141.578 | 62.231 | 67.352 | 141.426 | 62.686 | 67.847 | 141.630 |
106 | 60.840 | 66.301 | 141.103 | 59.273 | 67.658 | 141.619 | 62.532 | 67.707 | 141.578 | 62.007 | 67.004 | 141.227 | 62.686 | 67.847 | 141.630 |
107 | 60.871 | 66.318 | 141.107 | 59.368 | 67.660 | 141.619 | 62.537 | 67.712 | 141.580 | 62.858 | 67.987 | 141.676 | 62.686 | 67.848 | 141.630 |
108 | 61.128 | 66.464 | 141.145 | 60.091 | 67.676 | 141.619 | 62.571 | 67.750 | 141.597 | 62.698 | 67.858 | 141.634 | 62.686 | 67.848 | 141.630 |
109 | 62.042 | 67.140 | 141.349 | 61.872 | 67.757 | 141.623 | 62.651 | 67.822 | 141.623 | 62.687 | 67.848 | 141.631 | 62.686 | 67.848 | 141.630 |
1010 | 62.592 | 67.727 | 141.576 | 62.582 | 67.832 | 141.629 | 62.681 | 67.844 | 141.629 | 62.686 | 67.848 | 141.630 | 62.686 | 67.848 | 141.630 |
1011 | 62.676 | 67.835 | 141.624 | 62.675 | 67.846 | 141.630 | 62.685 | 67.847 | 141.630 | 62.685 | 67.847 | 141.630 | 62.686 | 67.848 | 141.630 |
1012 | 62.684 | 67.846 | 141.630 | 62.684 | 67.847 | 141.630 | 62.685 | 67.847 | 141.630 | 62.685 | 67.847 | 141.630 | 62.686 | 67.848 | 141.630 |